Evolution inclusions with Clarke subdifferential type in Hilbert space

In this paper we mainly consider the following evolution inclusion with Clarke subdifferential: (∗){−ẋ(t)∈∂cφ(x(t))+G(t,x(t))x(0)=x0 where ∂c is the Clarke subdifferential and t∈I=[0,T] is a nonempty, bounded closed interval in R+=[0,+∞); H is a separable Hilbert space; x0∈D(∂cφ)={x∈H:∂cφ(x)≠ϕ}. First, we introduce some hypotheses on φ, under which we prove that there is a unique solution of (∗) when G is a single function. Based on this result, we obtain an existence theorem for G being a u.s.c. or l.s.c multifunction. Then the existence of an extremal solution of (∗) and the relaxation theorem are studied. In the end, we present two examples of PDE to illustrate the application of our results.